ADDITION AND RESOLUTION OF VECTORS 11 dent, the components of r along any one of the axes must equal the sum of the corresponding components of the given vectors. Therefore (3) can be split into the following three separate equations. rtf = ay+btf+ctf + • - • , (4) rz = a, + b, + c, + ••-.. It was shown in § 11 that when two vectors are parallel the algebraic sum of their magnitudes equals the magni- Yl Fro. 12. tude of their resultant. This result may be extended to any number of parallel vectors. Therefore we can put the vector equations of (4) into the following algebraic forms: rv- av+lv+cy+ (5) Equations (5) determine r through the following relations: (6) (7) 9" rrr rx COS oti = COS C*2 ^ ........J cos «3= where «i, «2, and «3 are the angles r makes with the axes. 16. Multiplication and Division of a Vector by a Scalar.— When a vector is multiplied or divided by a scalar the result is a vector which has the same direction as the original vector. If, in the equation b = ma, m be a scalar then b has the same direction as a but its magnitude is m times that of a.